Approximating a common solution of extended split equality equilibrium and fixed point problems

Abstract

In this paper, we study an extension of the split equality equilibrium problem called the extended split equality equilibrium problem. We give an iterative algorithm for approximating a solution of extended split equality equilibrium and fixed point problems and obtained a strong convergence result in a real Hilbert space. We further applied our result to solve extended split equality monotone variational inclusion and equilibrium problems. The result of this paper complements and extends results on split equality equilibrium problems in the literature.

1 Introduction

Let C be a nonempty, closed and convex subset of a Hilbert space H and let $T:C\rightarrow C$ be a mapping. A point $x\in C$ is said to be a fixed point of T if $Tx =x$. Denote the set of fixed points of the mapping T by F(T).

Definition 1.1

( [36]) Let C be a nonempty, closed and convex subset of a Hilbert space H and let $\lambda \in (-\infty , 1).$ A mapping $T:C\rightarrow H$ with $F(T)\ne \emptyset$ is called $\lambda$-demimetric if for any $x\in C$ and $x^{*}\in F(T)$,

$$\begin{aligned} \langle x-x^{*},x-Tx\rangle \ge \frac{1-\lambda }{2}||x-Tx||^{2}, \end{aligned}$$

(1.1)

Clearly every $\lambda$-strictly pseudocontractive mapping T with $F(T)\ne \emptyset$ is a $\lambda$-demimetric mapping. Also, we recall that a mapping $T:C\rightarrow H$ is called $(\alpha ,\beta )$- generalised hybrid (see, [21]), if there exists $\alpha ,\beta \in {\mathbb {R}}$ such that

$$\begin{aligned}&\alpha ||Tx-Ty||^{2}+(1-\alpha )||x-Ty||^{2}\nonumber \\&\quad \le \beta ||Tx-y||^{2}+(1-\beta )||x-y||^{2} \end{aligned}$$

(1.2)

for all $x,y\in C$. It has been shown that the class of $(\alpha ,\beta )$- generalised hybrid mappings generalises the nonexpansive mappings [37], nonspreading mappings [23, 24] and the hybrid mappings [35]. Moreover, if T is an $(\alpha ,\beta )$- generalised hybrid mapping and $F(T)\ne \emptyset$ [37], we have that for $x\in C$ and $x^{*} \in F(T),$

$$\begin{aligned}&\alpha ||x^{*}-Tx||^{2}+(1-\alpha )||x^{*}-Tx||^{2}\nonumber \\&\quad \le \beta ||x^{*}-x||^{2}+(1-\beta )||x^{*}-x||^{2} \end{aligned}$$

(1.3)

and hence $||Tx-x^{*}||\le ||x-x^{*}||.$ Therefore, we have that

$$\begin{aligned} 2\langle x-x^{*},x-Tx\rangle \ge ||x-Tx||^{2} \end{aligned}$$

(1.4)

and thus

$$\begin{aligned} \langle x-x^{*},x-Tx\rangle \ge \dfrac{1-0}{2}||x-Tx||^{2}, \end{aligned}$$

(1.5)

which implies that every $(\alpha ,\beta )$- generalised hybrid mapping with $F(T)\ne \emptyset$ is 0-demimetric.

Let $f:C\times C\rightarrow {\mathbb {R}}$ be a bifunction, then the equilibrium problem associated with f and the set C is: find $x\in C$ such that

$$\begin{aligned} f(x,y)\ge 0,~~\forall y\in C. \end{aligned}$$

(1.6)

A point $x\in C$ that satisfies (1.6) is called an equilibrium point. We shall in this work denote the set of solutions of equilibrium problem (1.6) by EP(f, C). The equilibrium problem can be applied to solve problems from other fields such as physics and economics (see for example [34]). Also, many optimisation problems such as variational inequality, convex minimization, and Nash equilibrium problems can be transformed in the form of equilibrium problem (1.6).

The equilibrium problem because of its importance has attracted the interest of many mathematicians who have developed and studied numerous iterative algorithms for the approximation of solutions of equilibrium problems (see, [13]). Moreover, authors have also taken interest in the problems of finding a common element of fixed points of nonlinear mappings and the set of solutions of equilibriums, [1, 10, 15, 17]. The study of this kind of problem was inspired by certain problems arising from signal processing, network resource allocation, and image recovery results in mathematical models whose constraint can be expressed as fixed point problems and/or equilibrium problems [20, 29].

Let $C_{1}$ and $C_{2}$ be nonempty closed and convex subsets of real Hilbert spaces $H_{1}$ and $H_{2},$ respectively and let $A:H_{1}\rightarrow H_{2}$ be a bounded linear operator. The split feasibility problem (first introduced by Censor and Elfving [8], for modelling inverse problems in ${\mathbb {R}}^{n}$) is :

$$\begin{aligned} \mathrm{find}\, x \in C_{1}\,{such\,that}\,Ax\in C_{2}. \end{aligned}$$

(1.7)

The split feasibility problem has been considered as a veritable area of study because of its applications in signal processing, image reconstruction and intensity modulated therapy [6, 7].

Let $H_{1}$ and $H_{2}$ be real Hilbert spaces and let $S:H_{1}\rightarrow {\mathbb {R}}\cup \{+\infty \}$ and $g:H_{2}\rightarrow {\mathbb {R}}\cup \{+\infty \}$ be closed convex proper functions. Attouch et al. [4] introduced the following convex optimization problem:

$$\begin{aligned} \min \{S(x)+g(y)+\dfrac{\mu }{2}Q(x,y),~ x\in H_{1},~y\in H_{2}\}, \end{aligned}$$

(1.8)

where $Q:H_{1}\times H_{2}\rightarrow {\mathbb {R}}^+$ is a nonnnegative quadratic form which couples the two variables x and y,  and $\mu$ is a positive parameter. An example of the nonnnegative quadratic form Q(x, y) is $Q(x,y)=||Ax-Bx||_{H_{3}}^{2},$ where $A\in L(H_{1},H_{3})$ and $B\in L(H_{2},H_{3})$ are Bounded linear operators acting from $H_{1}$ to $H_{3}$ and from $H_{2}$ to $H_{3}$ respectively (see, [3]). The optimization problem (1.8) can be applied to solve problems from various areas such as decision science and game theory, partial differential equations and mechanics, and optimal control and approximation theory [2, 4, 27].

Inspired by this kind of problem considered by Attouch et el. [3] and the interest to cover many situations such as decomposition methods for PDEs, Moudafi [32] introduced the following split equality problem. Let $A:H_{1}\rightarrow H_{3}$, $B:H_{2}\rightarrow H_{3}$ be two bounded linear operators, let $C_{1}$ and $C_{2}$ be nonempty closed and convex subsets of real Hilbert spaces $H_{1}$ and $H_{2}$ respectively. Find

$$\begin{aligned} x\in C_{1}, y\in C_{2}\, \mathrm{such\,that}\,Ax=By. \end{aligned}$$

(1.9)

Clearly the split equality problem (1.9) is a generalization of the split feasibility problem (1.7). Moreover, the spilt equality (1.9) allows asymmetric and partial relations between the variables x and y. Moudafi [31], further considered the following split equality fixed point theorem. Let $A:H_{1}\rightarrow H_{3}$, $B:H_{2}\rightarrow H_{3}$ be two bounded linear operators, let $C_{1}$ and $C_{2}$ be nonempty closed and convex subsets of real Hilbert spaces $H_{1}$ and $H_{2},$ respectively. Let $S:C\rightarrow C$ and $T:Q\rightarrow Q$ be non linear operators. Find

$$\begin{aligned} x\in F(S), y\in F(T)\, \mathrm{such\, that} \, Ax=By. \end{aligned}$$

(1.10)

Moudafi and Al-Shemas [33], proposed an iterative method for solving (1.10) for firmly quasi-nonexpansive operators as follows:

$$\begin{aligned} \left\{ \begin{array}{lllll} &{} x_{n+1}=S(x_{n}-\gamma _n A^{*}(Ax_n-By_n)),\\ &{} y_{n+1}=T(y_{n}+\gamma _n A^{*}(Ax_n-By_n)), \forall n\ge 0, \end{array} \right. \end{aligned}$$

(1.11)

where $\gamma _n\in (\epsilon ,\dfrac{2}{\lambda _{A}+\lambda _{B}}-\epsilon )$, $\lambda _{A}$ and $\lambda _{B}$ stand for the spectral radius of $A^{*}A$ and $B^{*}B$, respectively. Several iterative algorithms have been developed for solving the split equality problems and split convex feasibility problems [14, 16, 18, 19, 22, 25, 28, 30, 33, 38, 39].

In 2008, Atouch et al. [3] extended the convex optimization problem (1.8) to the case of n variables. Precisely, they considered the following general convex optimization problem:

$$\begin{aligned}&\min \left\{ \sum _{i=1}^{n}f_{i}(x_i)\right. \nonumber \\&\left. +\dfrac{1}{2}\sum _{1\le i\le j\le n}Q_{ij}(x_{i},x_{j}),~x_i\in H_i, i\in \{1,2,\cdots ,n\}\right\} ,\nonumber \\ \end{aligned}$$

(1.12)

where for $i=1,2,\cdots ,n,$ $H_{i}$ is a real Hilbert spaces, $f_i:H_i\rightarrow {\mathbb {R}}\cup \{+\infty \}$ is convex, lower semicontinuous and proper functional, and $Q_{ij}:H_{i}\times H_{j}\rightarrow {\mathbb {R}}$ is a nonnnegative continuous quadratic form. Similar to (1.8), one can choose $Q_{ij}(x_i,x_j)=||A_ix_i-A_jx_j||^{2}_Z,$ where $A_{i}\in L(H_{i},Z)$ is a bounded linear operator mapping $H_{i}$ to Z.

Recently, Che et al. [9], inspired by the work of Atouch et al. [3], introduced the following extended split equality problem (ESEP). Let H be a real Hilbert space. For $i=1,2,\cdots ,n,$ let $C_{i}$ be a nonempty closed convex subset of real Hilbert spaces $H_{i}$ respectively and $A_{i}:H_{i}\rightarrow H$ be bounded linear operators. Find

$$\begin{aligned}&x_1\in C_1, x_2\in C_2,\cdots , x_n\in C_n \nonumber \\&\quad \text {such that } A_1 x_1=A_2 x_2=\cdots = A_nx_n. \end{aligned}$$

(1.13)

They proposed the following algorithm: Let $(x_{1,1}, x_{1,2},\cdots ,x_{1,n})\in H_{1}\times H_{2}\times \cdots \times H_{n}$ be arbitrary. Calculate the (k+1)-th iterate via the following formula

$$\begin{aligned} \left\{ \begin{array}{lllll} &{} w_{k}=\dfrac{\sum _{i=1}^{n}A_{i}x_{k,i}}{n},\\ &{} x_{k+1,1}=P_{C_{1}}(x_{k,1}-\gamma _k A_{1}^{*}(A_1x_{k,1}-w_{k})),\\ &{} x_{k+1,2}=P_{C_{2}}(x_{k,2}-\gamma _k A_{2}^{*}(A_2x_{k,2}-w_{k})),\\ &{}\vdots \\ &{} x_{k+1,n}=P_{C_{n}}(x_{k,n}-\gamma _k A_{n}^{*}(A_nx_{k,n}-w_{k})), \end{array} \right. \end{aligned}$$

(1.14)

where the stepsize $\gamma _{k}\in (\epsilon , \underset{1\le i\le n}{\min }\{\dfrac{1}{\lambda _{A_{i}}}\}-\epsilon ),$ and $\lambda _{A_{i}}$ stands for the spectral radius of $A^{*}_{i}A_{i}$. They obtained a weak convergence result.

In this paper, we study the following Extended Split Equality Fixed Point and Equilibrium Problems (ESEFPEP) which is to find

$$\begin{aligned}&x_{1}\in F(T_{1})\cap EP(f_{1},C_{1}), x_{2}\in F(T_{2})\nonumber \\&\quad \cap EP(f_{2},C_{2}),\cdots , x_{n}\in F(T_{n})\cap EP(f_{n},C_{n})\nonumber \\&\quad \text {such that } A_{1}x_{1}=A_{2}x_{2}= \cdots =A_{n}x_{n} \end{aligned}$$

(1.15)

where $T_{i}:C_{i}\rightarrow C_{i}~ (i=1,2,\cdots , n)$ are demimetric mappings. We shall denote the solution set of (1.15) by $\Omega .$

2 Preliminaries

Lemma 2.1

([36, 37]) Let H be a Hilbert space and let C be a nonempty, closed and convex subset of H . Let $\lambda \in (-\infty , 1)$ and let T be a $\lambda$ -demimetric mapping of C into C . Then F(T) is closed and convex.

Lemma 2.2

([13]) Let C be a nonempty closed convex subset of a Hilbert space H and let $f:C\times C\rightarrow {\mathbb {R}}$ be a bi-function satisfying the following conditions:

(A1):

$f(x, x) = 0\, \mathrm{for\,all}\,x \in C;$

(A2):

f is monotone, that is, $f(x, y) + f(y, x)\le 0$ for all $x, y \in C;$

(A3):

for each $x, y, z \in C,$

$$\begin{aligned} \limsup _{t\downarrow 0}f(tz+(1-t)x,y)\le f(x,y); \end{aligned}$$

(A4):

for each $x \in C, y \mapsto f(x, y)$ is convex and lower semi-continuous.

Let $r >0$ and $x\in H.$ Then, there exists $z \in C$ such that

$$\begin{aligned} f(z, y) +\dfrac{1}{r}\langle y-z,z-x\rangle \ge 0, \forall y\in C. \end{aligned}$$

Lemma 2.3

( [38]). Assume that $\{a_{n}\}$ is a sequence of nonnegative real numbers such that

$$\begin{aligned} a_{n+1}\le (1-\alpha _{n})a_{n}+\alpha _{n}\delta _{n}, n\ge 0, \end{aligned}$$

where $\{\alpha _n\}$ is a sequence in (0, 1) and $\{\delta _{n}\}$ is a sequence in ${\mathbb {R}}$ such that

• (i) $\sum _{n=1}^{\infty }\alpha _{n}=\infty ,$

• (ii) $\underset{n\rightarrow \infty }{\limsup }\delta _{n}\le 0$ or $\sum _{n=1}^{\infty }|\alpha _n\delta _n|<\infty .$

Then $\underset{n\rightarrow \infty }{\lim }a_n=0.$

Lemma 2.4

([13]) Let C be a nonempty closed convex subset of a Hilbert space H and let f be a bifunction of $C\times C$ into ${\mathbb {R}}$ satisfying $(A1)-(A4).$ For $r > 0$ and $x \in H$ , define a mapping $T_{r}^{f}:H\rightarrow C$ as follows:

$$\begin{aligned}&T^{f}_{r}(x)\nonumber \\&\quad =\{z\in C:f(z,y)+\dfrac{1}{r}\langle y-z,z-x\rangle \ge 0,~~ \forall y\in C\},\nonumber \\ \end{aligned}$$

(2.1)

for all $x \in H.$ Then the following hold:

• (i) $T^{f}_{r}$ is single-valued;

• (ii) $T^{f}_{r}$ is firmly non-expansive, that is, for any $x, y \in H,$

  $$\begin{aligned} ||T^{f}_{r}(x)-T^{f}_{r}(y)||^{2}\le \langle T^{f}_{r}(x)-T^{f}_{r}(y),x-y\rangle ; \end{aligned}$$

• (iii) $F(T^{f}_{r}) = EP(C,f), ~~\forall r > 0;$

• (iv) EP(C, f) is closed and convex.

Lemma 2.5

Let H be a real Hilbert space. Then the following hold:

• (a) $||x + y||^{2}\le ||y||^{2} + 2\langle x, x +y\rangle$ for all $x, y \in H;$

• (b) $||x -y ||^{2} = ||x||^{2} + ||y||^{2}- 2\langle x, y\rangle$ for all $x, y \in H.$

• (c) $||\alpha x +(1-\alpha )y||^{2}=\alpha ||x||^{2}+(1-\alpha )||y||^{2}-\alpha (1-\alpha )||x-y||^{2},$ for all $x,y\in H \, \mathrm{and }\, \alpha \in (0,1).$

Definition 2.6

Let H be a Hilbert space and let C be a nonempty, closed and convex subset of H. A mapping $T: C\rightarrow H$ is called demiclosed if, for a sequence $\{x_n\}$ in C such that $x_n \rightharpoonup w$ and $x_n -T(x_n)\rightarrow 0$, then $w =Tw$ holds.

3 Main Results

Theorem 3.1

Let H be a real Hilbert space. For $i=1,2,\cdots , n,$ let $C_{i}$ be a nonempty closed and convex subset of real Hilbert space $H_{i}$ and let $A_{i}:H_{i}\rightarrow H$ be a bounded linear operator. Let $T_{i}:C_{i}\rightarrow C_{i}$ be $\lambda _i$ -demimetric and demiclosed mappings and let $f_{i}:C_{i}\times C_{i}\rightarrow {\mathbb {R}}$ be bifunctions satisfying conditions $(A1)-(A4)$ such that $\Omega \ne \emptyset .$ Let $(x_{1,1},x_{1,2},\cdots , x_{1,n})\in H_{1}\times H_{2}\times \cdots \times H_{n}$ be arbitrary and let $u_{i}\in H_{i} (i=1,2\cdots n)$ be arbitrary but fixed. Let the sequence $\{(x_{k,1},x_{k,2},\cdots x_{k,n})\}$ be generated as follows:

$$\begin{aligned} \left\{ \begin{array}{lllll} &{} w_{k}=\dfrac{\sum _{i=1}^{n}A_{i}x_{k,i}}{n},\\ &{} y_{k,1}=T_{r_{k}}^{f_{1}}(x_{k,1}-\gamma _k A_{1}^{*}(A_1x_{k,1}-w_k)),\\ &{} x_{k+1,1} =\alpha _k u_1+(1-\alpha _k)[(1-\beta _k)y_{k,1}+\beta _k T_1y_{k,1}],\\ &{} y_{k,2}=T_{r_{k}}^{f_{2}}(x_{k,2}-\gamma _k A_{2}^{*}(A_2x_{k,2}-w_k)),\\ &{} x_{k+1,2} =\alpha _ku_2+(1-\alpha _k)[(1-\beta _k)y_{k,2}+\beta _k T_2y_{k,2}],\\ &{}\vdots \\ &{} y_{k,n}=T_{r_{k}}^{f_{n}}(x_{k,n}-\gamma _k A_{n}^{*}(A_nx_{k,n}-w_k)),\\ &{} x_{k+1,n} =\alpha _ku_n+(1-\alpha _k)[(1-\beta _k)y_{k,n}+\beta _k T_ny_{k,n}],~ k\ge 1. \ \end{array} \right. \end{aligned}$$

(3.1)

where $\gamma _{k}\in (\epsilon ,\underset{1\le i\le n}{\min }\{\dfrac{1}{\gamma _{A_{i}}}\}-\epsilon )$ and $\gamma _{A_{i}}$ stands for the spectral radius of $A^{*}_{i}A_{i}$ . Also, $\{\alpha _{k}\}$ and $\{\beta _{k}\}$ are sequences in (0, 1) satisfying the following conditions

• (i) $\lim _{k\rightarrow \infty }\alpha _{k}=0$, $\sum _{k=1}^{\infty }\alpha _{k}=\infty ;$

• (ii) $0<\underset{k\rightarrow \infty }{\liminf }\beta _{k}<\underset{k\rightarrow \infty }{\limsup }\beta _{k}<1-\lambda$, $\lambda :=\underset{1\le i\le n}{\max }\{\lambda _{i}\}.$

• (iii) $r_{k}\ge r>0.$

Then the sequence $\{(x_{k,1},x_{k,2},\cdots x_{k,n})\}$ converges strongly to $(z^{*}_{1},z^{*}_{2},\cdots ,z^{*}_{n})\in \Omega .$

Proof

Let $(z_1,z_2,\cdots ,z_n)\in \Omega ,$ and ${\bar{z}}= A_1z_1=A_2z_2=\cdots =A_nz_n.$ Since $T_i$, $i=1,2,\cdots , n$ is $\lambda _i$-demimetric, we have,

$$\begin{aligned}&||y_{k,i}-z_i +\beta _k(T_iy_{k,i}-y_{k,i})||^{2} \nonumber \\&\quad =||y_{k,i}-z_i||^{2}+\beta _k^2||T_iy_{k,i}-y_{k,i}||^{2}\nonumber \\&\qquad +2\beta _k\langle y_{k,i}-z_i,T_iy_{k,i}-y_{k,i}\rangle \nonumber \\&\quad \le ||y_{k,i}-z_i||^{2}+\beta _k^2||T_iy_{k,i}-y_{k,i}||^{2}\nonumber \\&\qquad -\beta _k(1-\lambda _{i})||y_{k,i}-T_iy_{k,i}||^{2}\nonumber \\&\quad = ||y_{k,i}-z_i||^{2}+\beta _k(\beta _k-(1-\lambda _{i}))||T_iy_{k,i}-y_{k,i}||^{2}\nonumber \\&\quad \le ||y_{k,i}-z_i||^{2}. \end{aligned}$$

(3.2)

Therefore, from (3.1) and (3.2), we obtain

$$\begin{aligned} ||x_{k+1,i}-z_i||^{2}=\,& {} ||\alpha _k u_i +(1-\alpha _k)[(1-\beta _k)y_{k,i}\nonumber \\&+\beta _k T_iy_{k,i}]-z_i||^{2}\nonumber \\=\,& {} \alpha _k||u_i-z_i||^{2}+(1-\alpha _k)||(1-\beta _{k})y_{k,i}\nonumber \\&+\beta _k T_iy_{k,i}-z_i||^{2}\nonumber \\&-\alpha _k(1-\alpha _k)||u_i-[(1-\beta _{k})y_{k,i}\nonumber \\&+\beta _k T_iy_{k,i}]||^{2}\nonumber \\\le & {} \alpha _k ||u_i-z_i||^{2}+(1-\alpha _k)||(1-\beta _{k})y_{k,i}\nonumber \\&+\beta _k T_iy_{k,i}-z_i||^{2}\nonumber \\\le & {} \alpha _k ||u_i-z_i||^{2}+ (1-\alpha _k)||y_{k,i}-z_i||^{2}. \end{aligned}$$

(3.3)

Also from (3.1), we get

$$\begin{aligned} ||y_{k,i}-z_{i}||^{2}=\,& {} ||T_{r_{k}}^{f_{i}}(x_{k,i}-\gamma _{k}A_{i}^{*}(A_{i}x_{k,i}-w_{k}))-z_{i}||^{2}\nonumber \\\le & {} ||x_{k,i}-z_{i}-\gamma _{k}A_{i}^{*}(A_{i}x_{k,i}-w_{k})||^{2}\nonumber \\=\,& {} ||x_{k,i}-z_{i}||^{2}+\gamma _{k}^{2}||A_{i}^{*}(A_{i}x_{k,i}-w_{k})||^{2}\nonumber \\&-2\gamma _{k}\langle x_{k,i}-z_{i},A_{i}^{*}(A_{i}x_{k,i}-w_{k})\rangle \nonumber \\= & {} ||x_{k,i}-z_{i}||^{2}+\gamma _{k}^{2}||A_{i}^{*}(A_{i}x_{k,i}-w_{k})||^{2}\nonumber \\&-2\gamma _{k}\langle A_{i}(x_{k,i}-z_{i}),A_{i}x_{k,i}-w_{k}\rangle \nonumber \\= & {} ||x_{k,i}-z_{i}||^{2}+\gamma _{k}^{2}||A_{i}^{*}(A_{i}x_{k,i}-w_{k})||^{2}\nonumber \\&+\gamma _{k}[-||A_{i}x_{k,i}-A_{i}z_{i}||^{2} \nonumber \\&-||A_{i}x_{k,i}-w_{k}||^{2}+||A_{i}z_{i}-w_{k}||^{2}]\nonumber \\\le & {} ||x_{k,i}-z_{i}||^{2}-\gamma _{k}(1-\gamma _{k}||A_{i}||^{2})||A_{i}x_{k,i}-w_{k}||^{2}\nonumber \\&-\gamma _{k}||A_{i}x_{k,i}-A_{i}z_{i}||^{2}+\gamma _{k}||A_{i}z_{i}-w_{k}||^{2}. \end{aligned}$$

(3.4)

Thus it follows from (3.3) and (3.4) that

$$\begin{aligned} ||x_{k+1,i}-z_{i}||^{2}\le & {} \alpha _{k}||u_{i}-z_{i}||^{2}\nonumber \\&+(1-\alpha _{k})\left[ ||x_{k,i}-z_{i}||^{2}-\gamma _{k}(1-\gamma _{k}||A_{i}||^{2})||A_{i}x_{k,i}-w_{k}||^{2}\right. \nonumber \\&\left. -\gamma _{k}||A_{i}x_{k,i}-A_{i}z_{i}||^{2} + \gamma _{k}\dfrac{\sum _{i=1}^{n}||{\bar{z}}-A_{i}x_{k,i}||^{2}}{n}\right] , \end{aligned}$$

(3.5)

which implies

$$\begin{aligned}&\sum _{i=1}^{n}||x_{k+1,i}-z_{i}||^{2} \nonumber \\&\quad \le \alpha _{k}\sum _{i=1}^{n}||u_{i}-z_{i}||^{2}+(1-\alpha _{k})[\sum _{i=1}^{n}||x_{k,i}-z_{i}||^{2}\nonumber \\&\qquad -\sum _{i=1}^{n}\gamma _{k}(1-\gamma _{k}||A_{i}||^{2})||A_{i}x_{k,i}-w_{k}||^{2}]. \end{aligned}$$

(3.6)

Hence

$$\begin{aligned}&\sum _{i=1}^{n}||x_{k+1,i}-z_{i}||^{2} \nonumber \\&\quad \le \alpha _{k}\sum _{i=1}^{n}||u_{i}-z_{i}||^{2}+(1-\alpha _{k})\sum _{i=1}^{n}||x_{k,i}-z_{i}||^{2}\nonumber \\&\quad \le \max \{\sum _{i=1}^{n}||u_{i}-z_{i}||^{2},\sum _{i=1}^{n}||x_{k,i}-z_{i}||^{2}\}\nonumber \\&\quad \vdots \nonumber \\&\quad \le \max \{\sum _{i=1}^{n}||u_{i}-z_{i}||^{2},\sum _{i=1}^{n}||x_{1,i}-z_{i}||^{2}\}. \end{aligned}$$

(3.7)

Therefore $\{\sum _{i=1}^{n}||x_{k,i}-z_{i}||^{2}\}$ is bounded, which implies $\{x_{k,i}\}$ is bounded for each $i=1,2,\cdots ,n.$ We now consider two cases to obtain the strong convergence.

Case 1. Assume that $\{\sum _{i=1}^{n}||x_{k,i}-z_{i}||^{2}\}$ is monotonically decreasing. Then it follows that $\{\sum _{i=1}^{n}||x_{k,i}-z_{i}||^{2}\}$ is convergent and $\sum _{i=1}^{n}||x_{k,i}-z_{i}||^{2}-\sum _{i=1}^{n}||x_{k+1,i}-z_{i}||^{2}\rightarrow 0,k\rightarrow \infty .$

Now from (3.6), we have

$$\begin{aligned}&(1-\alpha _{k})\sum _{i=1}^{n}\gamma _{k} (1-\gamma _{k}||A_{i}||^{2})||A_{i}x_{k,i}-w_{k}||^{2} \nonumber \\&\quad \le \alpha _{k}(\sum _{i=1}^{n}||u_{i}-z_{i}||^{2}-\sum _{i=1}^{n}||x_{k,i}-z_{i}||^{2})\nonumber \\&\qquad + \sum _{i=1}^{n}||x_{k,i}-z_{i}||^{2}-\sum _{i=1}^{n}||x_{k+1,i}-z_{i}||^{2}\rightarrow 0,k\rightarrow \infty . \end{aligned}$$

(3.8)

That is

$$\begin{aligned} \lim _{k\rightarrow \infty }(1-\alpha _{k})\sum _{i=1}^{n}\gamma _{k} (1-\gamma _{k}||A_{i}||^{2})||A_{i}x_{k,i}-w_{k}||^{2} =0, \end{aligned}$$

which implies

$$\begin{aligned} \lim _{k\rightarrow \infty }||A_{i}x_{k,i}-w_{k}||^{2} =0, \,\mathrm{\,for\,each}\, i=1,2,\cdots ,n. \end{aligned}$$

(3.9)

But from Lemma 2.4 (ii), we have

$$\begin{aligned} 0\le & {} -\langle y_{k,i}-z_{i},y_{k,i}-z_{i}\rangle \nonumber \\&+\langle y_{k,i}-z_{i},x_{k,i}-\gamma _{k}(A_{i}^{*}(A_{i}x_{k,i}-w_{k})-z_{i}\rangle \nonumber \\= & {} \langle y_{k,i}-z_{i},x_{k,i}-\gamma _{k}A_{i}^{*}(A_{i}x_{k,i}-w_{k})-y_{k,i}\rangle , \end{aligned}$$

(3.10)

which implies

$$\begin{aligned}&\langle y_{k,i}-x_{k,i},y_{k,i}-z_{i}\rangle \nonumber \\&\quad \le \gamma _{k}\langle A_{i}^{*}(A_{i}x_{k,i}-w_{k}),z_{i}-y_{k,i}\rangle \nonumber \\&\quad \le \gamma _{k}||A_{i}^{*}(A_{i}x_{k,i}-w_{k})||||z_{i}-y_{k,i}||. \end{aligned}$$

(3.11)

It then follows from Lemma 2.5(b), (3.3) and (3.11) that

$$\begin{aligned} ||y_{k,i}-x_{k,i}||^{2}= & {} ||x_{k,i}-z_{i}||^{2}-||y_{k,i}-z_{i}||^{2}\nonumber \\&+2\langle y_{k,i}-x_{k,i},y_{k,i}-z_{i}\rangle \nonumber \\\le & {} ||x_{k,i}-z_{i}||^{2}-(1-\alpha _{k})||y_{k,i}-z_{i}||^{2}\nonumber \\&+2\langle y_{k,i}-x_{k,i},y_{k,i}-z_{i}\rangle \nonumber \\\le & {} ||x_{k,i}-z_{i}||^{2}-||x_{k+1,i}-z_{i}||^{2}\nonumber \\&+\alpha _{k}||u_{i}-z_{i}||^{2}\nonumber \\&+2\langle y_{k,i}-x_{k,i},y_{k,i}-z_{i}\rangle \nonumber \\\le & {} ||x_{k,i}-z_{i}||^{2}-||x_{k+1,i}-z_{i}||^{2}\nonumber \\&+\alpha _{k}||u_{i}-z_{i}||^{2}\nonumber \\&+2\gamma _{k}||A_{i}^{*}(A_{i}x_{k,i}-w_{k})||||z_{i}-y_{k,i}||. \end{aligned}$$

(3.12)

Thus

$$\begin{aligned}&\sum _{i=1}^{n}||y_{k,i}-x_{k,i}||^{2}\nonumber \\&\quad \le \sum _{i=1}^{n}||x_{k,i}-z_{i}||^{2}\nonumber \\&\qquad -\sum _{i=1}^{n}||x_{k+1,i}-z_{i}||^{2}+\alpha _{k}\sum _{i=1}^{n}||u_{i}-z_{i}||^{2}\nonumber \\&\qquad +2\gamma _{k}\sum _{i=1}^{n}||A_{i}^{*}(A_{i}x_{k,i}-w_{k})||||z_{i}\nonumber \\&\qquad -y_{k,i}||\rightarrow 0,k\rightarrow \infty . \end{aligned}$$

(3.13)

Therefore,

$$\begin{aligned} \lim _{k\rightarrow \infty }||y_{k,i}-x_{k,i}||=0,~i=1,2,\cdots ,n. \end{aligned}$$

Again from Lemma 2.5 (c) and (3.2), we have

$$\begin{aligned} ||x_{k+1,i}-z_{i}||^{2}=\,& {} \alpha _{k}||u_{i}-z_{i}||^{2}\nonumber \\&+(1-\alpha _{k})||y_{k,i}-z_{i}+\beta _{k}(T_{i}y_{k,i}-y_{k,i})||^{2}\nonumber \\&-\alpha _{k}(1-\alpha _{k})||y_{k,i}\nonumber \\&+\beta _{k}(T_{i}y_{k,i}-y_{k,i})-u_{i}||^{2}\nonumber \\\le & {} \alpha _{k}||u_{i}-z_{i}||^{2}+(1-\alpha _{k})||y_{k,i}-z_{i}\nonumber \\&+\beta _{k}(T_{i}y_{k,i}-y_{k,i})||^{2}\nonumber \\\le & {} \alpha _{k}||u_{i}-z_{i}||^{2}+(1-\alpha _{k})||y_{k,i}-z_{i}||^{2}\nonumber \\&+(1-\alpha _{k})\beta _{k}(\beta _{k}-(1-\lambda ))||T_{i}y_{k,i}-y_{k,i}||^{2}.\nonumber \\ \end{aligned}$$

(3.14)

which implies

$$\begin{aligned}&(1-\alpha _{k})\beta _{k}((1-\lambda )-\beta _{k})||T_{i}y_{k,i}-y_{k,i}||^{2} \nonumber \\&\quad \le \alpha _{k}||u_{i}-z_{i}||^{2}+(1-\alpha _{k})||y_{k,i}-z_{i}||^{2}-||x_{k+1,i}-z_{i}||^{2}\nonumber \\&\quad \le \alpha _{k}[||u_{i}-z_{i}||^{2}-||x_{k,i}-z_{i}||^{2}]+(1-\alpha _{k})||y_{k,i}-x_{k,i}||^{2}\nonumber \\&\qquad +||x_{k,i}-z_{i}||^{2}-||x_{k+1,i}-z_{i}||^{2}\nonumber \\&\qquad +2||y_{k,i}-x_{k,i}||||x_{k,i}-z_{i}||. \end{aligned}$$

(3.15)

Therefore,

$$\begin{aligned}&(1-\alpha _{k})\beta _{k}((1-\lambda )-\beta _{k})\sum _{i=1}^{n}||T_{i}y_{k,i}-y_{k,i}||^{2}\nonumber \\&\quad \le \alpha _{k}\sum _{i=1}^{n} [||u_{i}-z_{i}||^{2}-||x_{k,i}-z_{i}||^{2}]\nonumber \\& \qquad +(1-\alpha _{k})\sum _{i=1}^{n}||y_{k,i}-x_{k,i}||^{2}\nonumber \\&\qquad +\sum _{i=1}^{n}||x_{k,i}-z_{i}||^{2}-\sum _{i=1}^{n}||x_{k+1,i}-z_{i}||^{2}\nonumber \\&\qquad +2\sum _{i=1}^{n}||y_{k,i}-x_{k,i}||||x_{k,i}-z_{i}||\rightarrow 0,k\rightarrow \infty , \end{aligned}$$

(3.16)

which implies

$$\begin{aligned} ||T_{i}y_{k,i}-y_{k,i}||\rightarrow 0,k\rightarrow \infty , \forall i=1,2,\cdots ,n. \end{aligned}$$

(3.17)

Again from (3.1), we have

$$\begin{aligned} ||x_{k+1,i}-y_{k,i}||\le & {} \alpha _{k}||u_{i}-y_{k,i}|| \nonumber \\&+(1-\alpha _{k})||(1-\beta _{k})(y_{k,i}-y_{k,i})\nonumber \\&+\beta _{k}(T_{i}y_{k,i}-y_{k,i})||\nonumber \\= & {} \alpha _{k}||u_{i}-y_{k,i}||\nonumber \\&+(1-\alpha _{k})\beta _{k}||T_{i}y_{k,i}-y_{k,i}||\rightarrow 0,k\rightarrow \infty .\nonumber \\ \end{aligned}$$

(3.18)

Now,

$$\begin{aligned} ||x_{k+1,i}-x_{k,i}||\le \,& {} ||x_{k+1,i}-y_{k,i}||\nonumber \\&+||y_{k,i}-x_{k,i}||\rightarrow 0,k\rightarrow \infty . \end{aligned}$$

(3.19)

Since $\{x_{k,i}\}$ is bounded for all $i=1,2,\cdots ,n$ there exists a subsequence $\{x_{k_{j},i}\}$ of $\{x_{k,i}\}$ for each $i=1,2,\cdots ,n$ such that $\{x_{k_{j},i}\}$ converges weakly to $z^{*}_{i}\in C_{i}.$ From the assumption that $T_{i} (i=1,2,\cdots ,n)$ is demiclosed, (3.13) and (3.17), we have that $z^{*}_{i}\in F(T_{i})$ for each $i=1,2,\cdots ,n.$

We now show that $z^{*}_{i}\in EP(f_{i},C_{i}), i=1,2,\cdots ,n.$ From $y_{k_{j},i}=T_{r_{k_{j}}}^{f_{i}}(x_{k_{j},i}-\gamma _{k_{j}}A_{i}^{*}(A_{i}x_{k_{j},i}-w_{k_{j}})),$ we have

$$\begin{aligned}&f_{i}(y_{k_{j},i},v_{i})+\dfrac{1}{r_{k_{j}}}\langle v_{i}-y_{k_{j},i},y_{k_{j},i}-x_{k_{j},i}\rangle \nonumber \\&\quad +\dfrac{1}{r_{k_{j}}}\langle v_{i}-y_{k_{j},i},\gamma _{k_{j}}A_{i}^{*}(A_{i}x_{k_{j},i}-w_{k_{j}}\rangle \nonumber \\&\quad \ge 0, \forall v_{i}\in C_{i}, i=1,2,\cdots ,n. \end{aligned}$$

(3.20)

It then follows from the monotonicity of $f_{i} (i=1,2,\cdots ,n),$ that

$$\begin{aligned}&\dfrac{1}{r_{k_{j}}}\langle v_{i}-y_{k_{j},i},y_{k_{j},i}-x_{k_{j},i}\rangle \nonumber \\&\quad +\dfrac{1}{r_{k_{j}}}\langle v_{i}-y_{k_{j},i},\gamma _{k_{j}}A_{i}^{*}(A_{i}x_{k_{j},i}-w_{k_{j}})\rangle \nonumber \\&\quad \ge f_{i}(v_{i},y_{k_{j},i}), \forall v_{i}\in C_{i}, i=1,2,\cdots ,n, \end{aligned}$$

(3.21)

and since $y_{k_{j},i}\rightharpoonup z^{*}_{i}, i=1,2,\cdots , n$ and A4 that $f_{i}(v_{i},z^{*}_{i})\le 0, \forall v_{i}\in C_{i}, i=1,2,\cdots ,n.$

For $v_{i}\in C_{i}$, let $y_{i,t}:=tv_{i}+(1-t)z^{*}_{i}$ for all $t\in (0,1).$ Then clearly $y_{i,t}\in C_{i}, i=1,2,\cdots ,n.$ Therefore, from A1 and A4, we have

$$\begin{aligned} 0= & {} f_{i}(y_{i,t},y_{i,t})\nonumber \\\le & {} tf_{i}(y_{i,t},v_{i})+(1-t)f_{i}(y_{i,t},z^{*}_{i})\nonumber \\\le & {} tf_{i}(y_{i,t},v_{i}), \end{aligned}$$

(3.22)

which yields $f_{i}(y_{i,t},v_{i})\ge 0.$

Thus from A3, we obtain $f_{i}(z^{*}_{i},v_{i})\ge 0, i=1,2,\cdots ,n.$

Let ${\bar{w}}=\dfrac{\sum _{i=1}^{n}A_{i}z^{*}_{i}}{n},$ then it follows from (3.9) and the lower semicontinuity of the square norm that for $i=1,2,\cdots , n,$

$$\begin{aligned} ||A_{i}z^{*}_{i}-{\bar{w}}||^{2}\le \liminf _{k\rightarrow \infty }||A_{i}x_{k,i}-w_{k}||^{2}=0. \end{aligned}$$

Hence $A_{i}z^{*}_{i}-{\bar{w}}=0$, which yields

$$\begin{aligned}&A_{2}z^{*}_{2}+A_{3}z^{*}_{3}+\cdots +A_{n}z^{*}_{n}=(n-1)A_{1}z^{*}_{1},\\&A_{1}z^{*}_{1}+A_{3}z^{*}_{3}+\cdots +A_{n}z^{*}_{n}=(n-1)A_{2}z^{*}_{2}\\&\vdots \\&A_{1}z^{*}_{1}+A_{2}z^{*}_{2}+\cdots +A_{n-1}z^{*}_{n-1}=(n-1)A_{n}z^{*}_{n}, \end{aligned}$$

and solving, we obtain $A_{1}z^{*}_{1}=A_{2}z^{*}_{2}=\cdots =A_{n}z^{*}_{n}.$ Therefore, $(z^{*}_{1},z^{*}_{2},\cdots ,z^{*}_{n})\in \Omega .$ We now obtain the strong convergence

$$\begin{aligned} ||x_{k+1,i}-z^{*}_{i}||^{2}= & {} ||\alpha _{k}u_{i}+(1-\alpha _{k})[(1-\beta _{k})y_{k,i}\nonumber \\&+\beta _{k}T_{i}y_{k,i}]-z^{*}_{i}||^{2}\nonumber \\\le & {} (1-\alpha _{k})^{2}||(1-\beta _{k})y_{k,i}+\beta _{k}T_{i}y_{k,i}-z^{*}_{i}||^{2}\nonumber \\&+2\alpha _{k}\langle u_{i}-z^{*}_{i}, x_{k+1,i}-z^{*}_{i}\rangle \nonumber \\\le & {} (1-\alpha _{k})^{2}||y_{k,i}-z^{*}_{i}||^{2} +2\alpha _{k}\langle u_{i}\nonumber \\&-z^{*}_{i}, x_{k+1,i}-z^{*}_{i}\rangle \nonumber \\\le & {} (1-\alpha _{k})||x_{k,i}-z^{*}_{i}||^{2}\nonumber \\&-(1-\alpha _{k})[\gamma _{k} (1-\gamma _{k}||A_{i}||^{2})||A_{i}x_{k,i}-w_{k}||^{2}\nonumber \\&-\gamma _{k}||A_{i}x_{k,i}-z^{*}_{i}||^{2}\nonumber \\&+\gamma _{k}\dfrac{\sum _{i=1}^{n}||{\bar{z}}-A_{i}x_{k,i}||^{2}}{n}]\nonumber \\&+2\alpha _{k}\langle u_{i}-z^{*}_{i}, x_{k+1,i}-z^{*}_{i}\rangle . \end{aligned}$$

(3.23)

Therefore,

$$\begin{aligned} \sum _{i=1}^{n}||x_{k+1,i}-z^{*}_{i}||^{2}\le & {} (1-\alpha _{k})\sum _{i=1}^{n}||x_{k,i}-z^{*}_{i}||^{2}\nonumber \\&+2\alpha _{k}\sum _{i=1}^{n}\langle u_{i}-z^{*}_{i}, x_{k+1,i}-z^{*}_{i}\rangle . \end{aligned}$$

(3.24)

Recall that in a real Hilbert space a sequence $\{x_n\}$ is said to converge weakly to a point $x\in H$ if $\langle x_n,y\rangle \rightarrow \langle x,y\rangle$ for all $y\in H.$ Now since $||x_{k,i}-x_{k+1,i}||\rightarrow 0$ and $x_{k,i}\rightharpoonup z_{i}^{*},$ we have that $x_{k+1,i}\rightharpoonup z_{i}^{*}.$ Therefore, for each $i=1,2,\cdots ,n$ we get

$$\begin{aligned}&\limsup _{k\rightarrow \infty }\langle u_i-z_{i}^{*},x_{k+1,i}-z_i\rangle \nonumber \\&\quad =\lim _{j}\langle u_i-z_{i}^{*},x_{k_{j},i}-z_{i}^{*}\rangle \nonumber \\&\quad = \langle u_i-z_{i}^{*},z_{i}^{*}-z_{i}^{*}\rangle =0. \end{aligned}$$

Thus it is easy to see that $\underset{k\rightarrow \infty }{\limsup }~ 2\sum _{i=1}^{n}\langle u_{i}-z^{*}_{i}, x_{k+1,i}-z^{*}_{i}\rangle =0,$ thus by Lemma 2.3, it follows that

$$\begin{aligned} \sum _{i=1}^{n}||x_{k+1,i}-z^{*}_{i}||^{2}\rightarrow 0,k\rightarrow \infty , \end{aligned}$$

which implies

$$\begin{aligned} ||x_{k,i}-z^{*}_{i}||\rightarrow 0, k\rightarrow \infty , \text{for each}\, i=1,2,\cdots ,n. \end{aligned}$$

That is

$$\begin{aligned} x_{k,i}\rightarrow z^{*}_{i}, i=1,2,\cdots ,n. \end{aligned}$$

Case 2: Asumme that $\{\sum _{i=1}^{n}||x_{k,i}-z_{i}||^{2}\}$ is not a monotonically decreasing sequence. Set $\Gamma _{k}=\sum _{i=1}^{n}||x_{k,i}-z_{i}||^{2}, \forall k\ge 1$ and let $\tau :{\mathbb {N}}\rightarrow {\mathbb {N}}$ be a mapping for all $k\ge k_{0}$ (for some $k_{0}$ large enough) by

$$\begin{aligned} \tau (k):=\max \{l\in {\mathbb {N}}:l\le k, \Gamma _{l}\le \Gamma _{l+1}\}. \end{aligned}$$

Clearly, $\tau$ is a nondecreasing sequence such that $\tau (n)\rightarrow \infty$ as $n\rightarrow \infty$ and

$$\begin{aligned} 0\le \Gamma _{\tau (k)}\le \Gamma _{\tau (k)+1}, \,for \,all \, k\, \ge k_{0}. \end{aligned}$$

After a similar conclusion from (3.17) and (3.19) respectively, it is easy to see that

$$\begin{aligned} ||T_{i}y_{\tau (k),i}-y_{\tau (k),i}||\rightarrow 0, k\rightarrow \infty , \end{aligned}$$

and

$$\begin{aligned} ||x_{\tau (k)+1,i}-x_{\tau (k),i}||\rightarrow 0, k\rightarrow \infty . \end{aligned}$$

Because $\{x_{\tau (k),i}\}$ is bounded for all $i=1,2,\cdots ,n$ there exists a subsequence of $\{x_{\tau (k),i}\}$, still denoted by $\{x_{\tau (k),i}\}$ which converges weakly to $z^{*}_{i}\in F(T_{i})$, for $i=1,2,\cdots ,n.$ Also $\underset{k\rightarrow \infty }{\limsup }2\sum _{i=1}^{n}\langle u_{i}-z^{*}_{i},x_{\tau (k)+1,i}-z^{*}_{i}\rangle =0.$

Also from (3.24), we have

$$\begin{aligned} \sum _{i=1}^{n}||x_{\tau (k)+1,i}-z^{*}_{i}||^{2}\le & {} (1-\alpha _{\tau (k)})\sum _{i=1}^{n}||x_{\tau (k),i}-z^{*}_{i}||^{2}\nonumber \\&+2\alpha _{\tau (k)}\sum _{i=1}^{n}\langle u_{i}-z^{*}_{i}, x_{\tau (k)+1,i}-z^{*}_{i}\rangle . \end{aligned}$$

(3.25)

which implies

$$\begin{aligned} \alpha _{\tau (k)}\sum _{i=1}^{n}||x_{\tau (k),i}-z^{*}_{i}||^{2}\le & {} \sum _{i=1}^{n}||x_{\tau (k),i}-z^{*}_{i}||^{2}\nonumber \\&-\sum _{i=1}^{n}||x_{\tau (k)+1,i}-z^{*}_{i}||^{2}\nonumber \\&+2\alpha _{\tau (k)}\sum _{i=1}^{n}\langle u_{i}-z^{*}_{i}, x_{\tau (k)+1,i}-z^{*}_{i}\rangle . \end{aligned}$$

(3.26)

That is

$$\begin{aligned} \sum _{i=1}^{n}||x_{\tau (k),i}-z^{*}_{i}||^{2}\le & {} 2\sum _{i=1}^{n}\langle u_{i}-z^{*}_{i}, x_{\tau (k)+1,i}-z^{*}_{i}\rangle . \end{aligned}$$

(3.27)

Therefore,

$$\begin{aligned} \sum _{i=1}^{n}||x_{\tau (k),i}-z^{*}_{i}||^{2}\rightarrow 0, k\rightarrow \infty , \end{aligned}$$

(3.28)

which implies

$$\begin{aligned} \lim _{k\rightarrow \infty }||x_{\tau (k)+1,i}-z^{*}_{i}||=0, \, \mathrm{for}\, i\,=\,1,2,\cdots , n. \end{aligned}$$

Furthermore, for $k\ge k_{0}$, it is easy to see that $\Gamma _{\tau (k)}\le \Gamma _{\tau (k)+1}$ if $k\ne \tau (k)$ (that is $\tau (k)<k$), because $\Gamma _{j}\ge \Gamma _{j+1}$ for $\tau (k)+1\le j\le k.$ As a consequence, we obtain for all $k\ge k_{0},$

$$\begin{aligned} 0\le \Gamma _{k}\le \max \{\Gamma _{\tau (k)},\Gamma _{\tau (k)+1}\}=\Gamma _{\tau (k)+1}. \end{aligned}$$

Hence, $\lim _{k\rightarrow \infty }\Gamma _{k}=0$, thus

$$\begin{aligned} \lim _{k\rightarrow \infty }\sum _{i=1}^{n}||x_{k,i}-z^{*}_{i}||=0, \end{aligned}$$

that is

$$\begin{aligned} ||x_{k,i}-z^{*}_{i}||\rightarrow 0, k\rightarrow \infty , i=1,2,\cdots ,n. \end{aligned}$$

Thus we conclude that $\{(x_{k,1},x_{k,2},\cdots ,x_{k,n})\}$ converges strongly to $(z^{*}_{1},z^{*}_{2},\cdots ,z^{*}_{n})\in \Omega .$ this completes the proof. $\square$

Corollary 3.2

Let H be a real Hilbert space. For $i=1,2,\cdots , n,$ let $C_{i}$ be a nonempty closed and convex subset of real Hilbert space $H_{i}$ and let $A_{i}:H_{i}\rightarrow H$ be bounded linear operators. For $\lambda _{i}\in (0,1),$ let $T_{i}:C_{i}\rightarrow C_{i}$ be $\lambda _i$ -strictly pseudocontractive mappings $(i=1,2,\cdots , n)$ and let $f_{i}:C_{i}\times C_{i}\rightarrow {\mathbb {R}}$ be bifunctions satisfying conditions $(A1)-(A4)$ such that $\Omega \ne \emptyset .$ Let $(x_{1,1},x_{1,2},\cdots , x_{1,n})\in H_{1}\times H_{2}\times \cdots \times H_{n}$ be arbitrary arbitrary and let $u_{i}\in H_{i} (i=1,2\cdots n)$ be arbitrary but fixed. Let the sequence $\{(x_{k,1},x_{k,2},\cdots x_{k,n})\}$ be generated as follows:

$$\begin{aligned} \left\{ \begin{array}{lllll} &{} w_{k}=\dfrac{\sum _{i=1}^{n}A_{i}x_{k,i}}{n},\\ &{} y_{k,1}=T_{r_{k}}^{f_{1}}(x_{k,1}-\gamma _k A_{1}^{*}(A_1x_{k,1}-w_k)),\\ &{} x_{k+1,1} =\alpha _k u_1+(1-\alpha _k)[(1-\beta _k)y_{k,1}+\beta _k T_1y_{k,1}],\\ &{} y_{k,2}=T_{r_{k}}^{f_{2}}(x_{k,2}-\gamma _k A_{2}^{*}(A_2x_{k,2}-w_k)),\\ &{} x_{k+1,2} =\alpha _ku_2+(1-\alpha _k)[(1-\beta _k)y_{k,2}+\beta _k T_2y_{k,2}],\\ &{}\vdots \\ &{} y_{k,n}=T_{r_{k}}^{f_{n}}(x_{k,n}-\gamma _k A_{n}^{*}(A_nx_{k,n}-w_k)),\\ &{} x_{k+1,n} =\alpha _ku_n+(1-\alpha _k)[(1-\beta _k)y_{k,n}+\beta _k T_ny_{k,n}], \, k\ge 1. \ \end{array} \right. \end{aligned}$$

(3.29)

where $\gamma _{k}\in (\epsilon ,\underset{1\le i\le n}{\min }\{\dfrac{1}{\gamma _{A_{i}}}\}-\epsilon )$ and $\gamma _{A_{i}}$ stands for the spectral radius of $A^{*}_{i}A_{i}$ . Also, $\{\alpha _{k}\}$ and $\{\beta _{k}\}$ are sequences in (0, 1) satisfying the following conditions

• (i) $\lim _{k\rightarrow \infty }\alpha _{k}=0$, $\sum _{k=1}^{\infty }\alpha _{k}=\infty ;$

• (ii) $0<\underset{k\rightarrow \infty }{\liminf }\beta _{k}<\underset{k\rightarrow \infty }{\limsup }\beta _{k}<1-\lambda$, $\lambda :=\underset{1\le i\le n}{\max }\{\lambda _{i}\}.$ Then the sequence $\{(x_{k,1},x_{k,2},\cdots x_{k,n})\}$ converges strongly to $(z^{*}_{1},z^{*}_{2},\cdots ,z^{*}_{n})\in \Omega .$

Corollary 3.3

Let H be a real Hilbert space. For $i=1,2,\cdots , n,$ let $C_{i}$ be a nonempty closed and convex subset of real Hilbert space $H_{i}$ and let $A_{i}:H_{i}\rightarrow H$ be bounded linear operators. For $\delta _{i},\eta _{i}\in {\mathbb {R}},$ let $T_{i}:C_{i}\rightarrow C_{i}$ be $(\delta _{i},\eta _{i})$ - generalised hybrid mappings $(i=1,2,\cdots , n)$ and let $f_{i}:C_{i}\times C_{i}\rightarrow {\mathbb {R}}$ be bifunctions satisfying conditions $(A1)-(A4)$ such that $\Omega \ne \emptyset .$ Let $(x_{1,1},x_{1,2},\cdots , x_{1,n})\in H_{1}\times H_{2}\times \cdots \times H_{n}$ be arbitrary arbitrary and let $u_{i}\in H_{i} (i=1,2\cdots n)$ be arbitrary but fixed. Let the sequence $\{(x_{k,1},x_{k,2},\cdots x_{k,n})\}$ be generated as follows:

$$\begin{aligned} \left\{ \begin{array}{lllll} &{} w_{k}=\dfrac{\sum _{i=1}^{n}A_{i}x_{k,i}}{n},\\ &{} y_{k,1}=T_{r_{k}}^{f_{1}}(x_{k,1}-\gamma _k A_{1}^{*}(A_1x_{k,1}-w_k)),\\ &{} x_{k+1,1} =\alpha _k u_1+(1-\alpha _k)[(1-\beta _k)y_{k,1}+\beta _k T_1y_{k,1}],\\ &{} y_{k,2}=T_{r_{k}}^{f_{2}}(x_{k,2}-\gamma _k A_{2}^{*}(A_2x_{k,2}-w_k)),\\ &{} x_{k+1,2} =\alpha _ku_2+(1-\alpha _k)[(1-\beta _k)y_{k,2}+\beta _k T_2y_{k,2}],\\ &{}\vdots \\ &{} y_{k,n}=T_{r_{k}}^{f_{n}}(x_{k,n}-\gamma _k A_{n}^{*}(A_nx_{k,n}-w_k)),\\ &{} x_{k+1,n} =\alpha _ku_n+(1-\alpha _k)[(1-\beta _k)y_{k,n}+\beta _k T_ny_{k,n}], \, k\ge 1. \ \end{array} \right. \end{aligned}$$

(3.30)

where $\gamma _{k}\in (\epsilon ,\underset{1\le i\le n}{\min }\{\dfrac{1}{\gamma _{A_{i}}}\}-\epsilon )$ and $\gamma _{A_{i}}$ stands for the spectral radius of $A^{*}_{i}A_{i}$ . Also, $\{\alpha _{k}\}$ and $\{\beta _{k}\}$ are sequences in (0, 1) satisfying the following conditions

• (i) $\lim _{k\rightarrow \infty }\alpha _{k}=0$, $\sum _{k=1}^{\infty }\alpha _{k}=\infty ;$

• (ii) $0<\underset{k\rightarrow \infty }{\liminf }\beta _{k}<\underset{k\rightarrow \infty }{\limsup }\beta _{k}<1.$ Then the sequence $\{(x_{k,1},x_{k,2},\cdots x_{k,n})\}$ converges strongly to $(z^{*}_{1},z^{*}_{2},\cdots ,z^{*}_{n})\in \Omega .$

4 Applications

Here, we apply the result of Theorem 3.1 to solve the following extended split equality monotone variational inclusion and equilibrium problems (ESEMVIEP) which is to find

$$\begin{aligned}&x_{1}\in (B_{1}+S_{1})^{-1}(0)\cap EP(f_{1},C_{1}), x_{2}\in (B_{2}+S_{2})^{-1}(0)\nonumber \\&\quad \cap EP(f_{2},C_{2}),\cdots , x_{n}\in (B_{n}+f_{n})^{-1}(0)\cap EP(f_{n},C_{n})\nonumber \\&\quad \text { such that } A_{1}x_{1}=A_{2}x_{2}= \cdots =A_{n}x_{n} \end{aligned}$$

(4.1)

where $B_{i}:H_{i}\rightarrow 2^{H_{i}}\, (i=1,2,\cdots , n)$ are maximal monotone mappings and $S_{i}:H_{i}\rightarrow H_{i}\, (i=1,2,\cdots , n)$ are $\alpha _{i}$-ism mappings. We shall denote the solution set of (4.1) by $\Omega _{1}.$

A mapping $S:H\rightarrow H$ is said to be $\alpha$-inverse strongly monotone ($\alpha$-ism), if there exists a constant $\alpha >0$ such that

$$\begin{aligned} \langle Sx-Sy, x-y\rangle \ge \alpha \Vert Sx-Sy\Vert ^{2},\, for \,all\,x,y\in H. \end{aligned}$$

A set valued mapping $B:H\rightarrow 2^{H}$ is called monotone if for all $x,y \in H$, with $u\in B(x)$ and $v\in B(y)$ then

$$\begin{aligned} \langle x-y, u-v\rangle \ge 0 \end{aligned}$$

and is maximal monotone if the graph of B denoted as G(B) is not properly contained in the graph of any other monotone mapping. We recall that for multivalued mapping B,

$$\begin{aligned} G(B)=\{(x,y):y\in B(x)\}. \end{aligned}$$

The resolvent operator $J_{\lambda }^{B}$ associated with B and $\rho >0$ is the mapping $J_{\rho }^{B}: H\rightarrow H$ defined by

$$\begin{aligned} J_{\rho }^{B}(x)=(I+\rho B)^{-1}(x), \,\,x\,\in H. \end{aligned}$$

(4.2)

The resolvent operator $J_{\rho }^{B}$ is single valued, nonexpansive and 1-inverse strongly monotone (for example see [5]). Moreover $0 \in B(x)+S(x)$ if and only if $x=J_{\rho }^{B}(I-\rho S)(x), \, for\,all\,\rho >0$ (see [26]). If S is $\alpha$-ism mapping with $0<\rho <2\alpha$, then $J_{\rho }^{B}(I-\rho f)$ is nonexpansive and $F(J_{\rho }^{B}(I-\rho S))$ is closed and convex.

Lemma 4.1

[11, 12] Let H be a Hilbert space and $T : H \rightarrow H$ a nonexpansive mapping, then for all $x, y \in H$,

$$\begin{aligned}&\langle (x - T x) - (y -T y), T y - T x\rangle \nonumber \\&\quad \le \frac{1}{2}||(T x - x) - (T y -y)||^2 \end{aligned}$$

(4.3)

and consequently if $y \in F(T )$ then

$$\begin{aligned} \langle x-Tx, x-y\rangle \le \frac{1}{2}||T x - x||^{2} . \end{aligned}$$

(4.4)

Theorem 4.2

Let H be a real Hilbert space. For $i=1,2,\cdots , n,$ let $C_{i}$ be a nonempty closed and convex subset of real Hilbert space $H_{i}$ and let $A_{i}:H_{i}\rightarrow H$ be bounded linear operators. Let $B_{i}:H_{i}\rightarrow 2^{B_{i}}$ be maximal monotone mappings, $S_{i}:H_{i}\rightarrow H_{i}$ be $\mu _{i}$ -ism mappings with $0<\rho <2\mu _{i}$ and let $f_{i}:C_{i}\times C_{i}\rightarrow {\mathbb {R}}$ be bifunctions satisfying conditions $(A1)-(A4)$ such that $\Omega _{1}\ne \emptyset .$ Let $(x_{1,1},x_{1,2},\cdots , x_{1,n})\in H_{1}\times H_{2}\times \cdots \times H_{n}$ be arbitrary and let $u_{i}\in H_{i} (i=1,2\cdots n)$ be arbitrary but fixed. Let the sequence $\{(x_{k,1},x_{k,2},\cdots x_{k,n})\}$ be generated as follows:

$$\begin{aligned} \left\{ \begin{array}{lllll} &{} w_{k}=\dfrac{\sum _{i=1}^{n}A_{i}x_{k,i}}{n},\\ &{} y_{k,i}=T_{r_{k}}^{f_{1}}(x_{k,1}-\gamma _k(A_1x_{k,1}-w_k)),\\ &{} x_{k+1,1} =\alpha _ku_1+(1-\alpha _k)[(1-\beta _k)y_{k,1}+\beta _k J_{\rho }^{B_{1}}(I-\rho S_{1}) y_{k,1}],\, k\, \ge\,1. \\ &{} y_{k,2}=T_{r_{k}}^{f_{2}}(x_{k,2}-\gamma _k(A_2x_{k,2}-w_k)),\\ &{} x_{k+1,2} =\alpha _ku_2+(1-\alpha _k)[(1-\beta _k)y_{k,2}+\beta _k J_{\rho }^{B_{2}}(I-\rho S_{2})y_{k,2}],\\ &{}\vdots \\ &{} y_{k,n}=T_{r_{k}}^{f_{n}}(x_{k,n}-\gamma _k(A_nx_{k,n}-w_k)),\\ &{} x_{k+1,n} =\alpha _ku_n+(1-\alpha _k)[(1-\beta _k)y_{k,n}+\beta _k J_{\rho }^{B_{n}}(I-\rho S_{n})y_{k,n}],\, k\, \ge \,1 \end{array} \right. \end{aligned}$$

(4.5)

where $\gamma _{k}\in (\epsilon ,\underset{1\le i\le n}{\min }\{\dfrac{1}{\gamma _{A_{i}}}\}-\epsilon )$ and $\gamma _{A_{i}}$ stands for the spectral radius of $A^{*}_{i}A_{i}$ . Also, $\{\alpha _{k}\}$ and $\{\beta _{k}\}$ are sequences in (0, 1) satisfying the following conditions

• (i) $\lim _{k\rightarrow \infty }\alpha _{k}=0$, $\sum _{k=1}^{\infty }\alpha _{k}=\infty ;$

• (ii) $0<\underset{k\rightarrow \infty }{\liminf }\beta _{k}<\underset{k\rightarrow \infty }{\limsup }\beta _{k}<1.$ Then the sequence $\{(x_{k,1},x_{k,2},\cdots x_{k,n})\}$ converges strongly to $({\bar{z}}_{1},{\bar{z}}_{2},\cdots ,{\bar{z}}_{n})\in \Omega _{1}.$

Proof

From the assumption $0<\rho <2\mu _{i}$, we have that $J_{\rho }^{B_{i}}(I-\rho S_{i})$ is nonexpansive for each $i=1,2,\cdots , n,$. Thus from Lemma 4.1 (4.4), we have that

$$\begin{aligned}&\langle x-J_{\rho }^{B_{i}}(I-\rho S_{i})x, x-y\rangle \nonumber \\&\quad \le \frac{1}{2}||J_{\rho }^{B_{i}}(I-\rho S_{i}) x - x||^{2}\nonumber \\&\quad =\frac{1-0}{2}||J_{\rho }^{B_{i}}(I-\rho S_{i}) x - x||^{2}, \end{aligned}$$

(4.6)

that is $J_{\rho }^{B_{i}}(I-\rho S_{i})$ for each $i=1,2,\cdots , n,$ is a $\lambda$-demimetric mapping with $\lambda =0$. Thus the proof follows from Theorem 3.1. $\square$